After President Obama visited their conference last week and took questions from them, Republicans have become very keen on institutionalizing the event the way it is in Britain, where the prime minister routinely takes questions from the opposition in Parliament.
It appears to me that Republicans are simply looking to save face from having believed their own propaganda about Obama's inability to speak without a teleprompter. Also, it's standard debating technique for underdogs to be elevated by being granted a debate with the leader. The debate makes both appear equal and the challenger has less to lose than the leader.
Nevertheless, I think Republicans are being short-sighted in their demands. Would they really have wanted the grossly inarticulate George W. Bush to have had to take questions regularly from Democrats in Congress? I think not.
Moreover, the dismal quality of presidential news conferences does not lead me to think that much useful information will be gleaned from question time with members of Congress. They will just ask "gotcha" questions or about obscure issues that are only of interest to them.